Question: Simplify the following expression: $y = \dfrac{-8x^2- 39x+5}{x + 5}$
Solution: First use factoring by grouping to factor the expression in the numerator. This expression is in the form ${A}x^2 + {B}x + {C}$ First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-8)}{(5)} &=& -40 \\ {a} + {b} &=& &=& {-39} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-40$ and add them together. Remember, since $-40$ is negative, one of the factors must be negative. The factors that add up to ${-39}$ will be your ${a}$ and ${b}$ When ${a}$ is ${1}$ and ${b}$ is ${-40}$ $ \begin{eqnarray} {ab} &=& ({1})({-40}) &=& -40 \\ {a} + {b} &=& {1} + {-40} &=& -39 \end{eqnarray} $ Next, rewrite the expression as $({A}x^2 + {a}x) + ({b}x + {C})$ $ ({-8}x^2 +{1}x) + ({-40}x +{5}) $ Factor out the common factors: $ x(-8x + 1) + 5(-8x + 1)$ Now factor out $(-8x + 1)$ $ (-8x + 1)(x + 5)$ The original expression can therefore be written: $ \dfrac{(-8x + 1)(x + 5)}{x + 5}$ We are dividing by $x + 5$ , so $x + 5 \neq 0$ Therefore, $x \neq -5$ This leaves us with $-8x + 1; x \neq -5$.